Using machine learning to distinguish between authentic and imitation Jackson Pollock poured paintings: A tile-driven approach to computer vision

Jackson Pollock’s abstract poured paintings are celebrated for their striking aesthetic qualities. They are also among the most financially valued and imitated artworks, making them vulnerable to high-profile controversies involving Pollock-like paintings of unknown origin. Given the increased employment of artificial intelligence applications across society, we investigate whether established machine learning techniques can be adopted by the art world to help detect imitation Pollocks. The low number of images compared to typical artificial intelligence projects presents a potential limitation for art-related applications. To address this limitation, we develop a machine learning strategy involving a novel image ingestion method which decomposes the images into sets of multi-scaled tiles. Leveraging the power of transfer learning, this approach distinguishes between authentic and imitation poured artworks with an accuracy of 98.9%. The machine also uses the multi-scaled tiles to generate novel visual aids and interpretational parameters which together facilitate comparisons between the machine’s results and traditional investigations of Pollock’s artistic style.


Introduction
In 1952, the Abstract Expressionist Jackson Pollock poured fluid paint onto a vast canvas rolled out across his studio floor and created his masterpiece, Blue Poles: Number 11, 1952 [1].The painting represents the culmination of 10 years of developing his 'pouring' technique and the 'all-over' style that it generated.In contrast to conventional brush contact with the canvas surface, the constant stream of paint produced continuous trajectories that wove together into a uniform pattern that lacked conventional compositional values-no center of focus, no up or down, and no left or right.
As artistic recognition for his revolutionary style grew and the commercial value of his work soared, judgments of authenticity became increasingly crucial.A number of damaging controversies have plagued the Pollock world, fueled by painting prices that exceed $100M and by the growing number of fakes appearing on the art market [2,3].If a poured painting of unknown origin is found today, how could we determine with reliability whether it is a masterpiece or a fake?In addition to the staggering financial consequences, rigorous methods are needed to protect the legacies of our most treasured artists.
Previous Pollock controversies were escalated by incorrect attributions made by wellrespected Pollock experts, highlighting the challenge that the visual complexity of Pollock's images presents for the human eye.This challenge forms the central scientific question of our study.If we replace the human observer with an artificial intelligence (AI) machine, what level of machine accuracy (MA) could be achieved when classifying the complexity of poured artworks into Pollock and non-Pollock categories?A MA close to 100% would suggest that artificial intelligence can distinguish Pollock's artistic signatures more readily than some of the best Pollock scholars.Although the capability of AI machines to out-perform humans is not unusual-indeed, this ability fuels many current applications across society-it presents a unique dilemma for the art world.Pollock's work was created for appreciation by humans and not machines.Perhaps the machine's superior ability to distinguish between masterpieces and imitations represents AI's version of art appreciation?
A high MA would also resolve a debate that has troubled the Pollock world from the moment he started to create his unusual patterns.Although many art theory essays have celebrated Pollock's unique talent, the fact that Pollock scholars can sometimes fail to spot fakes fuels the public perception that his work is no more than an inevitable consequence of pouring paint-and that a lay person could readily match Pollock's artistic achievements.Although the AI machine can't judge aesthetic worth, a high MA value would provide objective and quantitative proof that Pollock's work is a unique form of artistic expression.
In recent years, a variety of AI techniques have been applied to a growing number of other artists.An obvious approach for paintings featuring faces is to use AI-powered facial recognition methods.Focusing on the Renaissance master Raphael, this strategy identified a 97 percent similarity between the face of the Virgin Mary depicted in his confirmed painting Sistine Madonna and the face in the disputed work de Brecy Tondo [4].In contrast to training on the 'form' of the painted images (e.g.faces, figures,etc), an alternative strategy is to focus on the tell-tale painting techniques used by artists to generate the images.An example study examined more than 80,000 individual brushstrokes by Picasso, Matisse, and Schiele and classified them with an accuracy of up to 90% [5].Similarly, a team known as Art Recognition focused on artistic techniques such as brushwork and use of color, along with object placement within the canvas and other compositional characteristics.Their recent projects include confirming the authenticity of a van Gogh self-portrait and determining that an alleged painting by Max Perchstien is in fact by the infamous forger Wolfgang Beltrach [6].Devoid of the illustrative content of traditional art and constructed from splatters rather than careful brushwork, what would be the optimal AI approach for Pollock's work?
The authors formed an art-science collaboration called Art Intelligence to quantify MA performance as a way of gauging AI's potential to 'understand' Pollock's work.To promote the acceptance of this AI approach within the art world, we focus on well-established rather than novel machine learning models.Models will be more influential when resolving future Pollock controversies if they have a robust track record for addressing crucial applications across society.For the models considered in our study, these applications include analyzing faces at airports to maintain our national security and scanning medical images to ensure our health [7,8].
Accordingly, we train our machine to learn the visual characteristics of poured artworks using an artificial neural network called ResNet [9] and by exploiting an approach called Transfer Learning.ResNet employs 'deep' networks consisting of many layers and nodes ('neurons') that process information hierarchically.By employing ResNet's many hierarchical layers, the machine can detect a vast array of everyday visual signatures.We complement this machine architecture by reserving the last few layers of our neural network for discriminating between artworks by Pollock and those not by Pollock.We do this by showing the network the largest ever assembled digital collection of Pollock artworks, imitations of Pollock artworks, and a variety of other abstract artworks.We compare the performance of various versions of Resnet to other established machine models (including AlexNet, DenseNet, and SqueezeNet) and more recently developed Vision Transformers (including Pyramid Vision Transformer, Swin Transformer, and Multi-Axis Vision Transformer).Ultimately, we focus on Resnet50 based on its superior MA and proven track record across a variety of applications [7,9,10].
Whereas the chosen model architecture is deliberately traditional, the novel aspects of our study focus on the input (image ingestion) and output (interpretational parameters) stages of the process.In terms of input into the machine, we emphasize that although our collection of 588 works is comprehensive it is nevertheless significantly smaller than data sets typically used for machine learning.For example, cosmologists apply AI to thousands of images of the night sky [11].However, these same limits apply to human inspection of Pollock's work.We therefore anticipate that AI will provide a valuable step forward in bringing state-of-the-art scientific techniques for spotting and quantifying patterns in complex data to the art world.The machine's ability to perform these tasks can be expected to improve substantially if novel strategies can lift the restrictions presented by the limited image sets.How can the data input be boosted in order to achieve a high MA for Pollock's work?This forms the second scientific question of our study.
To accommodate the small number of images, our machine learns about the artworks by dividing each work into an array of tiles.This image ingestion method exploits 2 key characteristics of Pollock's all-over style.Firstly, because of the spatial uniformity of the all-over style, tiles at different locations are expected to have similar visual features to each other.Secondly, previous investigations examined the occurrence of fractal patterns [12] in the all-over style .We therefore use many tile sizes at each location.Because of the scale invariance of fractals, these multi-scaled tiles are expected to share visual features.In effect, each of Pollock's artworks features multiple 'sub-Pollocks' found at different locations and at different size scales-each displaying Pollock's tell-tale signature.Our fractal tiling approach therefore aligns with the fractal character of the images and by doing so boosts the amount of art images that we can use during the training of our machine.In total, a data set of 97,275 Pollock tiles and 150,242 non-Pollock tiles is employed in our process.
By integrating this novel ingestion method into our current model, our machine distinguishes between Pollock poured paintings and artworks created by other artists with a MA of 98.9%.When the machine is presented with an artwork of unknown origin, it generates a Pollock Matching Factor (PMF) to quantify the degree to which the artwork captures the visual appearance of Pollock's work using a scale from 0 to 1. Using our current model, we find that artworks with PMF values that reach or exceed the PMF 'Threshold' of 0.56 can be considered a close match to this visual appearance.This Threshold is set at the PMF that maximizes the MA for distinguishing between the Pollock and non-Pollock works.The machine gains confidence in the visual match to Pollocks as the PMF increases beyond the Threshold and similarly gains confidence in a miss-match as the PMF decreases below the Threshold.
Although the 'black box' quality of artificial neural networks highlights their immunity to human influence, this same quality limits insights into the visual signatures that the machine is employing.Our third scientific question therefore focuses on developing strategies to understand the machine's output.In particular, can our AI approach provide a new 'eye' on Pollock's all-over style and how it developed over a decade of refinement?As with the input strategy, the innovative step relies on our employment of multi-scaled tiles.We use these tiles to generate Pollock 'Maps' that examine the artistic signatures at different locations across the artwork and Pollock 'Zoom-in charts' that examine the signatures at different size scales.These visual aids are coupled with 4 parameters that allow us to quantify the key aspects of Pollock's revolutionary style.These provide an interdisciplinary bridge between our new scientific approach and the wealth of ideas presented in traditional art theory studies of Pollock's work.We show that fractality and spatial uniformity are important signatures used by our machine when identifying Pollocks.In particular, we show a strong correlation between PMF and the scale invariance of Pollock's signatures, indicating that fractal detection is prominent in the machine's decisions.
Given the prevalence of fractals in nature's scenery (for example, trees, clouds, mountains, rivers, and coastlines are all shaped by fractals [12]), our machine provides an objective connection between Pollock's poured paintings and nature.Frequently referred to as being natural in appearance, his complex patterns can be seen as a direct distillation of nature's geometry onto his canvases, supporting the description of 'Fractal Expressionism' [35].Although our parameters are developed for art interpretation of Pollocks, analogous approaches could readily be employed in other interdisciplinary studies of complex data aimed at establishing connections between traditional and novel interpretations of a subject.In particular, our novel tiling approach and its associated interpretive tools are well-suited for AI analysis of fractal patterns.Given the prevalence of fractals in nature, potential future fundamental and applied investigations include medical imaging of fractals in the body (e.g.tumors, neurons, veins, bronchial trees, bone fissures) [36], meteorology (e.g.clouds, rain, lightning, wind patterns) [12], geography (e.g.trees, plants, rivers, coastlines, mountains, mud cracks, rock textures) [12], and astronomy (e.g.dark matter distributions, star clusters, moon craters) [37][38][39].Artificial physical systems could also benefit from our AI analysis.This includes investigations of the fractal distributions of transport routes [40].
We discuss our novel input and output approaches relative to other techniques to emphasize their distinct characteristics.Finally, we emphasize that our technique is not intended to be applied in isolation when used for attributing a poured work to Pollock.The results will be most useful for authenticity studies when coupled with other important information such as materials analysis, provenance, and connoisseurship.Notably, we expect our AI inspection to complement the visual inspections of Pollock experts in a unique marriage between computer and human vision.Inevitably, this marriage will be tested by the growing prevalence of AI-generated art.We show that our current AI technique can detect current AI-generated Pollock imitations.As techniques develop, it will be fascinating to see if the ability to detect Pollock imitations continues to outpace the ability to generate them.Perhaps future AI machines will become so proficient that they can not only replicate Pollocks but develop his style in new directions based on subtle career developments spotted by the machine.Within this context, we emphasize that our current study is simply the preliminary step to many future studies of abstract artworks.

Background
Fig 1 highlights the challenge faced by authenticity studies by showing Pollock's Blue Poles side by side with a known imitation.Investigations aimed at distinguishing real from imitation artworks need to probe beyond the superficial similarities displayed by these artworks.To do so, investigations typically integrate artistic and scientific approaches.From the sciences, a range of analysis techniques can be employed to determine the material composition and date of the paint, the canvas, and the frame.From the arts, provenance studies assess the painting's history while connoisseurs conduct visual comparisons with established paintings from the artist's catalog (in Pollock's case, the Catalogue Raissonne [41]) to determine if the 'hand' of the artist is present.
For many artists, this combination of research tools yields powerful evidence for attributing paintings.Unfortunately, Pollock's history adds substantial confusion to these approaches.Pollock is known to have bartered some paintings and so authentic paintings might exist for which the provenance is long lost.Pollock was also the subject of a film at his career peak.This inspired a wave of imitation poured paintings that date from Pollock's era and are composed from similar materials.Because of these factors, visual inspection forms the central cornerstone of Pollock investigations.However, when faced with the staggering commercial implications coupled with the pressures of international media attention, subjective judgments have become increasingly difficult to obtain and defend against potential litigation.This creates an art world vulnerable to the acceptance of fakes and to the denial of long-lost treasures.
In response, one of this paper's authors (Taylor) pioneered the use of computer analysis to supplement the visual experts' inspection with quantitative and objective pattern detection techniques [13].Based on frequent references to the "organic" and "natural" appearance of Pollock's all-over style, along with Pollock's declaration that "my concerns are with the rhythms of nature" [1], this computer analysis focused on fractals.The analysis quantified a key visual element of Pollock's work-the fractal complexity generated by the patterns that repeat at increasingly fine size scales [12].Hints of their telltale pattern repetition had already been noticed in Pollock's era by art critics, journalists, and Pollock himself: "knit together of a complicity of identical and similar elements" (Clement Greenberg) [1], "The large in Pollock is an accumulation of the small "(T.J. Clark) [1], "patterns all roughly similar in character . . .over the surface of the picture" (William Rubin) [1], "Pollock is as strong from a distance as he is close to" (Alfred Frankenstein) [1], and "my paintings didn't have any beginning or end" (Pollock) [1].
Commencing with Taylor's initial publication in 1999 [13], 13 groups have since used various forms of fractal analysis to explore Pollock's artistic signature .In each case, computers were used to investigate similarities in the statistical characteristics of the painted patterns occurring across different size scales.In particular, the fractal dimensions (D) of Pollock's poured patterns have been employed to distinguish his artistic signatures from those of his imitators [17].Two groups have also used fractal techniques to create computer-generated imitation Pollocks [26,33].Subsequent neuroscience experiments highlighted the aesthetic impact of fractals [42,43].In particular, demonstrations of the shared visual qualities of nature's and Pollock's fractals inspired the Fractal Expressionism art movement [35].The scholarly investigation of Pollock's fractals informed a high-profile Pollock authentication investigation in 2005 when the Pollock-Krasner Foundation collaborated with Taylor to determine the origin of the 'Matter Collection [2,20].Fractal analysis was applied to 6 of 32 newly discovered poured paintings.These were found to differ in fractal characteristics [17] from 14 established Pollocks, consistent with subsequent pigment analysis showing that some of the paints dated from after Pollock's death.In another investigation of the 'Knoedler Collection' [3] by the International Foundation for Art Research, fractal analysis again highlighted fractal differences between one of the paintings and established Pollocks.This was consistent with the later discovery that the paintings were created by a contemporary artist.Although fractals have therefore played a useful role in authenticity studies, Taylor emphasized that they are one of many visual characteristics needed to capture the rich experience of viewing Pollock's paintings [17].Identifying a more comprehensive set of Pollock's visual characteristics would lend power to the process of separating his masterpieces from their imitations.
Could a computer train itself to identify and learn from a set of pattern characteristics in an artwork?Although asking a computer to make decisions about $100M paintings might at first seem to place too much trust in a novel technique, AI is already being employed to perform critical functions for society.In parallel, AI is experiencing a growing role in generating, classifying, and examining the authenticity of diverse works across the arts (for a recent review, see [44]).Prominent teams using AI for authenticity studies of paintings include Artrendex and Art Recognition [6,45].Table 1 presents some further examples presented in chronological order [46][47][48][49][50][51][52].
In the early 2010s, attempts at art authentication such as those of Qi et al [46] relied on more traditional computer vision techniques coupled with Fisher Information scores to categorize artworks.Because their research focused on the brushstrokes of Impressionist and Post-Impressionist art, their approach might not be expected to transfer well to the task of identifying Pollock's poured paintings.Nonetheless, in 2015 Shamir employed a similar technique to discriminate between Pollock forgeries and Pollock originals using a Fisher Information score after extracting a set of 3000 visual features [47].Despite the small data set of 26 Pollock poured works, his machine learning technique achieved a MA of 93%.Interestingly, we note that Shamir found that the fractal parameters in the set were the dominant distinguishing tools used by the machine.
By 2015 a crucial shift was underway in the computer vision world in the form of the deep learning capabilities of artificial neural networks.Convolutional neural networks (CNNs) were becoming popular and, significantly, the paper detailing Resnet was published at this time [9].CNNs make use of a limited set of "filters" which are then convolved mathematically across an image.By reducing the input layer to a small number of filters, CNNs perform well at approximating the visual characteristics of animals, and out-perform traditional neural networks Another form of deep learning-Vision Transformers (ViTs)-drives the latest trends in computer vision.ViTs 'tokenize' images in order to digest their information.This image tokenization is similar to how the transformers in large-language models tokenize words and sentences to digest text.In both cases, transformers decompose the data sets into smaller chunks and then extract information from the chunks [54].Last year, Schaerf et al demonstrated the effectiveness of ViTs for identifying art-works by Vincent van Gogh.
Although Pollock's abstract artworks have been the focus of high profile authenticity controversies involving staggering financial and artistic consequences, they have not yet benefited from an AI deep learning analysis.Given ResNet's established history in art classification and other applications, we focus our model comparisons on ResNet architectures along with a number of other CNNs.We also include several ViTs because of their emerging status in AI applications.
Considering our novel image ingestion approach, we note that our integration of tiling into the ResNet model is similar in some ways to the tokenizing approach utilized by ViTs.However, our tiling strategy differs in one key aspect: when our technique decomposes the images into tiles, we deliberately discard their relative positions while ViTs retain this information.Our tiling strategy is based on the fractal composition of Pollock's all-over style in which each tile serves as an independent Pollock image.As such, our study uses deep learning networks in a manner consistent with the previous scientific studies indicating that fractal parameters are useful descriptions of Pollock's complex patterns.Training our neural network on an array of multi-scaled tiles is therefore an intriguing solution to boosting image numbers while aligning our investigations with previous quantitative approaches to Pollock's artistic signatures.Our unique tiling approach, and its alignment with fractal patterning, also establishes the novelty of our interpretative Zoom-in Charts and Maps.For example, whereas other AI approaches identify the spatial locations of interesting features (e.g., unusual brushstrokes in paintings [6]), our Maps instead compare the visual signatures of well-defined regions of the canvas at different locations and size scales.

Image sets
The images of the 588 artworks used in our study were acquired in collaboration with The Pollock-Krasner Foundation, The Pollock-Krasner Study Center, The International Foundation for Art Research, and Francis V. O'Connor (chief Pollock connoisseur and co-author of the Catalogue Raissonne).The collection and analysis method of all images complies with the terms and conditions for the sources of the data.The S1 Pollock poured artwork.This category features all 189 Pollock artworks that satisfy our definition of poured works and that have known images.Pollock used a variety of painting techniques across his career [1].Our definition of pouring is broad and is based on artworks that feature patterns generated by fluid liquid poured onto the artwork's surface.To ensure our technique is robust to variations in image quality and color variations, we use multiple images of each painting when possible.More specifically, we employ 2 collections: one set features color images typically with high image resolution (scanned at 1200 px/inch from a variety of high quality art books [55][56][57][58][59][60][61][62][63][64][65][66][67]) and the other set features grayscale images with lower but still good image resolution (scanned at 600 px/inch from the Pollock's Catalogue Raissonne [41]).Using this approach, 118 of the 189 Pollock artworks feature images from both sets.
Non-Pollock art works.This second category comprises 2 groups of artwork.The first group features a diverse range of 284 poured works created by other artists.These include artworks generated specifically for this authenticity project (e.g.32 adult and 18 children's paintings created under the controlled conditions of 'Dripfest' events [17,31], and paintings generated by mechanical devices such as the Pollockizer [35]), 100 commercially-available poured artworks, poured artworks by well-known artists (e.g. by Michael Baldwin, Max Ernst, Sam Francis, Arshile Gorky, Hans Hofmann, Henri Michaux, Norman Rockwell, Niki de Saint Phalle, etc) and established poured imitations (e.g from the Knoedler Collection, the Matter Collection, by Mike Bidlo, Ed Harris, and Francis O'Connor).To add robustness, the non-Pollock category also features a diverse group of 115 abstract 'non-poured' abstract works by famous artists (e.g. by George Braque, Jasper Johns, Wassily Kandinski, Paul Klee, Willem De Kooning, Joan Miro, Clyfford Still, etc) [68].Only 1 image of each artwork is used in the non-Pollock category.The majority of the images are scanned at 1200 px/inch from a variety of high quality art books, prints, and transparencies.Photographed works (Dripfest works, One by the Pollockizer, Untitled by the Wind Machine, and Dummy by Richard Taylor) use high resolution settings (200-400 px/inch).Images of the commercial poured paintings are downloaded from the internet (72 px/inch).

Image pre-processing
Prior to training, images undergo a refined cropping process that employs a semi-automatic technique.Selecting the 4 corners of the painting manually, we then determine the minimum rectangle that encapsulates all of these points.We crop the image to this minimum rectangle.We tile each cropped image by covering the image with an array of identical squares and we then crop these to create a set of square sub-canvases.Based on the fractal model, we repeat this tiling process for various tile sizes, starting from squares with 10cm side lengths (the smallest tile width is set at 10 cm to avoid image resolution effects) and increasing in 5cm increments up to the maximum square size allowable for the painting.We center the tiling process on the image.As an example, for an image of size 265cm x 265cm we start the tiling process 2.5cm in from the left edge and 2.5cm down from the top edge.We continue until we reach the last tile, which is 2.5cm in from the right edge and 2.5cm up from the bottom edge.Once completed, we re-size each tile to a standardized 256 x 256 pixels and feed them into the neural network for classification.To illustrate this process, consider Blue Poles.We tile its 212.1cm × 488.9cm canvas with squares with side lengths ranging from 10cm up to 210cm, resulting in 2,387 tiles.Notably, our approach refrains from introducing any additional augmentations to the images beyond this pre-processing step.

Data partitioning
Machine learning typically requires a set of images to train the machine (the 'training set') and a second set to iteratively evaluate the machine's performance (the 'validation set').Traditionally, the validation set is used to evaluate machine models featuring different hyperparameters.In addition, we reserve a 'hold out' set of images that ensures that the iterative process of evaluating models with different hyper-parameters is not prone to any unseen biases.In other words, we ensure that the model will generalize to images that it has never seen.This is especially important for our circumstances whereby the set of known images of Pollock's artworks is not anticipated to grow significantly.The hold out set differs from the validation set in that it is never used to make decisions for improving or tuning our model.We evaluate the performance of the different models using only the validation set.Once we decide on the final model, we then test it on the hold out set.We refer to the combination of the validation and hold out sets as the 'inference set'.
Out of our 189 Pollock poured works, 142 (75.1%) are selected for the training set, 33 (17.5%) for the validation set, and 14 (7.4%) for the holdout set.For the Pollock images, the holdout set is randomly selected from the 189 paintings with the exception of deliberately including Blue Poles (as one of his most iconic images) and at least 1 painting from each year that he created a poured painting (i.e.1943 and 1946-1955).Out of our 399 non-Pollock works, 304 (76.2%) are selected for the training set, 72 (18.0%) for the validation set, and 23 (5.8%) for the holdout set.
Importantly, all of the tiles extracted from an image in our training set are used during the model training process.None of the tiles extracted from an image in our training set are used during model validation or when testing the model using the holdout set.The same is true for both the holdout and validation images.In total, a dataset of 97,275 Pollock tiles and 150,242 non-Pollock tiles are collected.Table 2 summarizes the relative numbers of images dedicated to each role.

Training
The machine uses the tiles described above to learn about the visual features of the artworks.The machine compares all of the tiles within the established Pollocks to the tiles within works known to be created by other artists.During this comparison, the machine identifies a collection of visual features that are useful for distinguishing between the Pollock and non-Pollock artworks.Based on the features present in a tile, it then assigns a value from 0 and 1 to the tile, which we call the Pollock Signature.A Pollock Signature of 0 indicates that the tile does not have the distinguishing visual features of a Pollock work, while a value of 1 indicates that it does.Values closer to 1 indicate increasing levels of confidence from the machine that the artwork contains the distinguishing visual features of a Pollock.All tiles are considered equally in training.The

Data quantification and visualization
The Pollock Dial.Because the PMF values vary from 0 to 1, we can visualize this information on a dial to easily compare PMFs of different artworks.The Pollock Dial plots the PMF of artworks using angular position with the PMF increasing in the clockwise direction.The Dial focuses on images from the inference set.Using the images on the Pollock Dial, we identify a PMF 'Threshold'-artworks that reach or exceed the PMF 'Threshold' can be considered a close match to the visual appearance of Pollock's works.This Threshold is calculated by determining the PMF that maximizes the MA when distinguishing between the collections of Pollock and non-Pollock works.
The Zoom-in charts and scaling parameters.The Zoom-In Charts examine how the Pollock Signatures vary with tile size.To generate the Charts, the machine groups tiles by their size and calculates the average Pollock Signature for each tile size.This average value is then plotted as a function of tile size using a bar chart.The Chart also displays the PMF of the whole artwork (which corresponds to the average across the tile sizes of all of their average Pollock Signatures).Bars with Pollock Signatures below the PMF Threshold are colored red while those above are green.
As discussed in the Introduction, previous research employed computers to perform a fractal analysis of Pollock's poured paintings.For fractal artworks, the statistical qualities of the painted patterns repeat at increasingly fine size scales.To be consistent with fractals, we expect the Pollock Signatures to similarly repeat at different tile sizes.We introduce Scale Invariance (SI) to quantify the variation in the Pollock Signature with tile size.This is based on the rootmean-square variation across all tile sizes.Its range is normalized using the function f(y) = 1 -(2*y).Adopting this normalization, SI = 1 corresponds to no variation in Pollock Signature as we zoom in and SI = 0 corresponds to the maximum possible variation.We also introduce Magnification (M) to quantify the extent of the zoom-in, with M representing the ratio of the sizes of the largest and smallest tile widths.
The signature maps and spatial parameters.Whereas the Pollock Zoom-Ins examine the variation in Pollock Signatures with size scale, the Signature Maps examine how the Pollock Signature varies for different locations across the artwork.We generate a Map for each painting in the inference set.At each location, the Map averages the Pollock Signature of the various sized tiles at that location.In other words, at each pixel location we identify all of the multi-scaled tiles that contain this pixel.The Pollock Signatures of these tiles are then averaged.This value is plotted at the pixel location using color coding (red for values below the PMF Threshold and green for values above).During this process, we slide the arrays of tiles across the image and calculate the average of the Pollock Signatures across the various slide positions.This sliding technique generates a smooth map by eliminating any discontinuities caused by crossing the boundaries of tiles with large Signature differences.The completed Map allows us to identify regions of the painting that deviate away from Pollock's style.We emphasize that, although related to the work's PMF, these Map Signatures use a different averaging technique to the one used to generate the PMF.
We introduce 2 parameters to quantify spatial variations in the Map.Uniformity (U) quantifies the variation in the Pollock Signature across different locations.This is based on the root-mean-square variations of the Pollock Signatures in the width and height directions.The range of the variation in each direction is then normalized using the function f(y) = 1 -(2*y).Adopting this normalization, U = 1 corresponds to no variation and U = 0 corresponds to the maximum possible variation.The U value for the painting corresponds to the mean of the U values for the 2 directions.Coverage (C) employs a pixel count to quantify the relative sizes of the red and green coverage in the Map, with C = 0 corresponding to all locations being red and C = 1 corresponding to all regions being green.
Fractal analysis: Differential box counting and shift differential box counting.Some previous forms of fractal analysis of Pollock's poured works extracted layers with different colors from the painting and then analyzed the pattern scaling properties of each of these layers separately [17].Other approaches investigated the scaling properties of the 'whole' painting (ie. the combination of all of the layers in the painting).This is usually done by examining the luminance variations of the patterns.Because we train the machine on the 'whole' pattern, we adopt the luminance approach and apply Differential Box Counting (DBC) and Shift Differential Box Counting (Shift-DBC) algorithms to calculate the works' fractal dimension D. These techniques are commonly used in the study of natural and synthetic patterns with fractal-like characteristics.They allow researchers to quantify and classify the structure of images from various disciplines, including geology, biology, and computer vision [69].
For the DBC technique, a square crop of the grayscale image of the artwork is converted into a 3D representation by plotting the pixel brightness in the height direction (normalized to the painting's physical dimensions).This 3D space is covered with an array of identical 'boxes', each with a side-length L. At each location in the square crop, we then calculate the number of boxes that fit into the difference between the minimum and maximum grayscale values in the height direction.NB is the sum of these box counts across all of the locations.This calculation is repeated for arrays of boxes with different L values.A scaling plot is then generated which plots NB vs L on log-log scales.Fractal behavior is characterized by the power law relationship, NB ~L−D , and D can therefore be extracted from the slope of the linear part of the scaling plot [69].To convert this dimension in 3D space to the painting's physical 2D space, a value of 1 is subtracted from the D value obtained from the scaling plot.
The Shift-DBC technique is an extension of DBC technique and introduces box shifting.Instead of placing boxes in a fixed grid, Shift-DBC shifts the box positions, allowing a more accurate estimation of D. The D value obtained from these 2 methods provides insight into how the image's features repeat at different scales to generate the overall fractal pattern.In particular, because higher D values correspond to steeper slopes in the scaling graph, high-D images generate larger NB values at smaller scales than corresponding lower-D images.In this way, D quantifies the relative contributions of the fine and coarse patterns in the fractal image.It therefore serves to quantify the visual complexity of fractal patterns such as those found in Pollock's works.D lies on a scale between 1 and 2 with simple, sparse fractals lying closer to 1 and rich, intricate fractals lying closer to 2.

Model selection and performance
We train our machine using the model architecture ResNet, which is an abbreviation for Residual Network.This artificial neural network was introduced by researchers at Microsoft Research in 2015 and soon after won a general image classification challenge [9].It is recognized as a powerful tool used frequently in computer vision tasks, particularly for recognizing objects in images.Artificial neural networks such as ResNet consist of many hierarchical layers analogous to those used by our brain when processing images.By employing ResNet, the machine can detect a vast array of everyday visual signatures.We complement this model architecture by reserving the last few layers of our neural network for discriminating specifically between artworks by Pollock and those not by Pollock.We do this by showing the network our digital collection of Pollock artworks, imitations of Pollock art works, and a variety of other abstract artworks.In this way, our machine transcends from the equivalent of an art novice to a Pollock expert.
More specifically, ResNet's innovation over other neural networks lies in its introduction of 'skip connections' that help the network to learn and optimize more effectively [10].Transfer Learning (LT) leverages knowledge gained from one task to improve performance for a different but related task.Instead of starting from scratch, TL uses a pre-trained model that has already learned basic features from a larger and more varied data set.We use TL to adapt ResNet from its task of classifying images featured in the Imagenet data set to our specific Pollock-related task [70].Imagenet-1K provides images of a variety of objects, ranging from beagles to violins [71].This pre-training saves time and computational resources, exposes the machine to a rich set of visual features, and improves the machine's ability to generalize beyond a specific type of image.The initial layers of the pre-trained model are used as a foundation.The later layers are adjusted to fit our Pollock-detection task.For our project, we utilize fastai [72] and timm [73] model libraries for neural network architectures.
A list of hyperparameters that we employ during model development can be found in the S2 Table .MA is a crucial parameter for assessing model performance.Another criterion is minimizing the PMF difference between images of the same painting from different sources.The following hyper-parameters are also important to consider: learning rate, batch size, and number of epochs.Learning rate determines how fast a model learns from the data.A higher rate produces faster learning but runs the risk of overshooting without settling on a good solution, while a lower rate produces slower learning but benefits from being more precise.The batch rate is the number of tiles used in each iteration of training.The number of epochs is the number of times the entire training data set is processed by the model.We choose a learning rate of 10 −3 , a batch size of 64, and 1 epoch.We note that the high accuracy and performance is not increased by considering different epochs or batch sizes (see S1 Fig).
In  capabilities of our chosen model.We find that the ViTs trained without our tiling approach consistently underperform compared to the CNNs.The purpose of training ViTs using our tiling method is to more directly compare their performances to the CNN architectures.Since ViTs already 'tokenize' an image into smaller chunks, their native data ingestion mechanism might be expected to reduce the impact of applying our tiling approach for these models.However, incorporating our tiling into ViTs nevertheless improves their MA performance.Taken together, these results highlight the suitability of using ResNet50 coupled with our tiling ingestion method for classifying the all-over style of images in our data set.
For all the CNN model architectures trained in  .For training the red ViTs, we use a larger image size (512 x 512 x 3), 1 epoch, and a reduced 'batch size' of 4 because of computational limitations.We note that, because tiny_vit is a more computationally efficient ViT architecture, we are able to run models with a batch size of 32 and 2 epochs (after which there is no improvement in MA).This still results in an inferior performance (faded red top) when compared to ResNet50.
Ultimately, we choose ResNet50 as our main model architecture over the other CNN architectures due to its consistently high MA and its established track record for a variety of image classification tasks [7,9,11,74].We note that ResNet50 exhibits comparable performance to ResNet101 while requiring significantly fewer computational resources.We observe a notable improvement in model performance when using ResNet50 compared to ResNet34 with only a marginal increase in training time.We choose ResNet50 over the ViTs considered due to ResNet50's performance in Fig 3 and because its longer history of operation across society might be more compelling for the current art world.
We note that ResNet50's performance deteriorates if we exclude our tiling process and simply train on the whole image: MA drops by 16.7% and 9.4% for batch sizes 4 and 64, respectively.To reach this higher performance, we explore different 'voting' schemes in terms of how the Pollock Signatures of each of the individual tiles would be used when calculating the PMF.Two example schemes include assigning each tile size a single 'vote' and giving each tile (irrespective of size) a single vote.The former approach yields a higher MA.Due to the prevalence of smaller tile sizes, the latter approach would be vulnerable to a dominance of small-scale characteristics.We also investigate giving different tile sizes a weighted 'vote' but we keep with the simpler equal voting scheme because the weighted approach doesn't improve performance.
Finally, in terms of selecting which images are utilized in the validation set, we employ multiple random combinations to identify an effective validation set.This ensures robustness across different sets of images.While training on various validation sets yields good performance overall, we ultimately select the validation set that delivers the best results for ResNet50.This comprehensive approach allows us to fine-tune our model selection and achieve optimal MA when distinguishing between the Pollock and non-Pollock groups.

Model generalization
In order to determine if our technique can generalize to other artists, we perform the following analysis.Rather than attempting to identify artworks by Pollock, we select another artist category from our collection of images and instead determine if our model can identify this category.We choose the Dripfest images for this test.These are a set of paintings created by 18 children and 32 adults in controlled settings.An example of each is shown in Fig 4 .These novice painters attempted to imitate Pollock by pouring paint.We apply the same approach as before, but instead train our model to identify Dripfest paintings.To do this, we randomly sample a variety of images from our image collection, including authentic works by Pollock.This ensures a diverse selection of non-Dripfest paintings while maintaining a balanced training set.
One big difference from classifying Pollock artworks focuses on the number of paintings.Whereas we had 189 works in the Pollock category, the Dripfest category features 50.We therefore sub-sampled our data set of non-Dripfest images to achieve a balance of 'imitation' and 'authentic' paintings.Importantly, whereas approximately 50% of the paintings in our main study are by Pollock, only 16% of the images in our generalization test are by Pollock.Table 3 shows the image partitioning in more detail.Despite this much smaller number of images used in training, our machine performs exceptionally well with this alternative data configuration.Utilizing this new data set, we achieve a MA of 100% by using a Threshold for the Matching Factor (i.e the equivalent of PMF) of 0.85.This high accuracy is consistent with previous fractal investigations which successfully differentiated Dripfest paintings from Pollocks [17,31].As we will see, our machine uses fractal content predominantly in its decisions.This result demonstrates that our model can be applied to other artists employing the pouring style.(Pollock), Free Form (Pollock), One (Pollockizer), Untitled (Henri Michaux), Enchanted Forest (Pollock), and Untitled Mural (Pollock).The PMF Threshold is shown by the dashed radial line.We find that setting the Threshold at 0.56 generates the highest machine accuracy (MA = 98.9%) for distinguishing between the Pollock and non-Pollock groups.The image of a demonstration painting, Dummy (1504 x 112.8cm, by Richard Taylor), is shown at the Dial's center (a full image is shown later in Fig 13).Dummy is selected from the non-Pollock validation set and serves as a useful demonstration because its midrange PMF (0.55, represented by the black arrow in the inner circle) lies close to the Threshold.

Pollock matching factors and The Pollock Dial
The Pollock artworks with the lowest PMFs on the Dial are 2 of his earliest poured paintings-Water Birds from 1943 (0.56) and Free Form from 1945 (0.60).The majority of works by Pollock have the highest PMF value of 1.The non-Pollock artworks on the Dial have PMFs that span the range from 0 to 0.95.The Dial features only 2 images that the machine miss-classifies as Pollocks (i.e. that meet or exceed the Threshold).One was generated by 'The Pollockizer'-a chaotic pendulum developed to generate Pollock imitations [35] (see Discussion).The other was Untitled by Henri Michaux (see Discussion).Whereas the Dial focuses on images in the inference set, we might also compare PMFs of paintings that are used in the training set when helpful.S4 Table provides a list of PMF values for all of the Pollock paintings.
Because the 2 groups of artwork congregate mainly at the 2 extremes of the Dial (non-Pollocks at low PMF and Pollocks at high PMFs), many of the data points are superimposed.Because of this, in Fig 6 we show a histogram of the numbers of artworks, n, at various PMF values for the 2 groups.In this figure, we also show how the machine accuracy decreases as the PMF Threshold is moved away from its optimal value of 0.56.Because of the relatively few paintings with mid-PMF values, the accuracy remains fairly stable for mid-values and then deteriorates significantly at the very low and high values.In future research, we plan to calculate a machine confidence to express the probability of artworks having Pollock's signatures based on PMF value.This will address the discrete character of the Threshold.However, it is clear from the histogram's 2 distributions that the machine gradually gains confidence in a visual match to Pollock's work as the PMF increases beyond the Threshold and then gains very high confidence close to PMF = 1.Similarly, the machine gradually gains confidence in a visual miss-match as the PMF decreases below the Threshold and then gains very high confidence close to PMF = 0. Image quality confirmation.To increase robustness to variations in image quality and color variation, we use 2 sets of Pollock images: high quality color images and lower quality grayscale images (see Methods for more details).All images tested from both sets are classified correctly, with a mean PMF difference between the 2 sets of only 0.01.This small difference confirms that the technique is robust to reasonable variations in image quality.We also convert the color images to grayscale and find that the average PMF drops by only 0.02.This small difference indicates that the Pollock Signatures used by the machine are predominantly influenced by the patterns rather than the colors of the Pollocks' features.
In Fig 7, we show the deterioration in PMF that occurs as we artificially reduce the image resolution for the high resolution, color images and lower resolution, grayscale images.We plot the Resolution Fraction (RF) on the horizontal axis.RF = 1 corresponds to the scanned resolution whereas, for example, RF = 1/30 corresponds to decreasing the pixel resolution along each direction (width and height) by a factor of 30.When the colored lines drop below the horizontal line (indicating the PMF Threshold of 0.56), we refer to this as falling off the

The all-over style and pollock timeline
We declared 3 scientific goals in the Introduction.The first and second are linked: can we achieve a high MA for distinguishing Pollocked poured works by integrating a robust, established machine model with a novel image ingestion approach based on multi-scaled tiles?Our resulting high value, MA = 98.9%, demonstrates the power of machine learning for future Pollock authenticity studies when its results are combined with other techniques such as human vision inspection, provenance investigations, and materials analysis.The high MA also provides scientific evidence that Pollock's artistic signature is quantifiably different to those of In addition to the machine parameters, we also chart changes in Pollock's productivity (N is the number of paintings per year) and composition (A is the canvas area and AR is its aspect ratio).In each case, the black dots represent individual paintings (many of which are superimposed) and the red lines represent the average values for each year.
The colored backgrounds represent the phases of his development from an art history perspective: 'preliminary' (pink), 'transitional' (purple), 'classic' (green), and 'final' (yellow) [1].His 'preliminary' works are composed of a few poured trajectories superimposed on a dominant brush-worked background.His transitional works feature a growing contribution of interacting poured trajectories dominating over the brushwork.His 'classic' works are characterized mainly by poured trajectories.His 'final' works mark a fall off from his large, densely poured masterworks of 1950-52.Intriguingly, there is little variation in PMF with year.Aside from some very early variations, his distinguishing characteristics remain constant through the years.This implies that, although the variations between the 4 phases are significant for charting Pollock's visual development, they all display Pollock's 'hand'.The differences in Pollock's evolution are subtle when compared to the differences between Pollocks and non-Pollocks.
The circled value to the right of each panel in Fig 16 quantifies the correlation between the PMF and each of the parameters, where 0 corresponds to no correlation (i.e. the relationship is random) and 1 corresponds to the strongest correlation.Interestingly, there is little correlation between PMF and most parameters (N, A, AR, M).This indicates that Pollock's Signature remains strong as these characteristics vary across the paintings.Notably, Pollock's Signature has no correlation with the canvas shape and size that he chose to work on.Taken together, this implies that the presence of the canvas edge has little impact on his Signature.This is consistent with traditional descriptions of his painting process.These picture Pollock as painting beyond the confines of his canvas, with the studio floor capturing the paint trajectories that extend beyond the artwork.There is also only a very weak correlation with the number of paintings he created at a given time.This is intriguing because at this peak in 1950 Pollock created some paintings in parallel rather than consecutively.There has been speculation concerning the impact of this switching between works on their appearance.Our result suggests that his Signature is strong whether it arose from his peak production or from rarer works painted in isolation.
To examine the correlation with the 4 machine parameters in detail,

Pollock's all-over style
Although Pollock is famous for the spatial uniformity of his all-over style, this quality of his work hasn't previously undergone a detailed examination using quantitative image analysis techniques.The above results support the traditional picture of his all-over style.The mean values of C and U are both close to 1, indicating that expansive regions of the canvas surface have very strong Signatures (colored green in the Maps) and, accordingly, there is little variation in these Signatures across different locations.Our Tests involving changes in canvas orientation are further consistent with his all-over style.The average PMF values at different orientations (rotations along with vertical and horizontal flips of the canvas) of the Pollocks in the inference set do not vary from the average PMF measured in the original orientation.
The Pollock Maps shown in Figs 14 and 15 emphasize their value as interpretational tools.Blue Poles serves as a powerful demonstration of the spatially-uniform Signature of his all-over style.In particular, the 8 poles are sufficiently splattered that they do not impact the uniformity of the Signatures nor the overall PMF.This effect is seen in other examples of his work.For example, Comet features a long white line stretching from top to bottom, and The Deep features a large dark region at its center.These features do not disturb the associated Maps because of their splattered character.More extreme deviations, such as the cut-out shape in Fig 15, result in a darkening of the map in the impacted region.However, the high Signatures in the surrounding regions ensure a high PMF.This is seen in other examples of his work.For example, The Wooden Horse features a wooden object that darkens the impacted region but doesn't dip the PMF below the Threshold.and shown in the bottom bar).The right column shows relative colors to highlight small changes.For Water Birds, the Signature varies from 0.607 (purple) to 0.632 (yellow), Blue Poles varies from 0.997 to 0.998, and Untitled varies from 0.897 to 0.986.These values emphasize the subtle relationship between PMF and 'all-overness': Untitled has a higher PMF than Water Birds, even though it has a larger radial variation in its Signature.
Intriguingly, all 3 paintings display a systematic deterioration in Signature as the canvas edge is approached.This indicates that even if Pollock continued to paint beyond the artwork's intended boundary, the boundary nevertheless had a subtle effect on his pouring technique.There are clues that Pollock sometimes spotted this deterioration and took measures to hide it -in particular, for some works he folded parts of the painted canvas behind the work to highlight the central region.Clearly, this wasn't necessary for Blue Poles, where our results show the deterioration is minimal.This deterioration effect is consistent with our decision to use the poured patterns on Pollock's studio floor (i.e. the heavily deteriorated pattern formed by the trajectories that completely missed the canvas) as a non-Pollock pattern in the training process (see later).Whereas the balancing motions of his body are proposed to generate one set of fractals dominating at size scales above the transitional size of approximately 5cm, the spattering of the fluid paint is proposed to generate a second set of fractals at smaller scales [17].Because our machine process focuses on tile sizes starting at 10cm, it should be sensitive to Pollock Signatures generated by the balancing motions.However, we note that for several Pollocks (these include Number 22 (1950), Number 24 (1950), Brown and Silver II (1951), and Untitled (1952)) we do see that the Pollock Signatures at the 10cm tile size are smaller than those for the larger tile sizes, suggesting that perhaps the spattering process might have started to dominate at the 10cm tile size for these paintings.
In addition to considering the degree of scale invariance of the patterns, fractal studies also look at scaling parameters (such as fractal dimension D) that quantify the rate at which the patterns shrink with magnification [12].The fractal model of Pollock's work introduced distinct fractal dimensions to quantify the scaling behavior of the 2 size regimes-DD for the 'drip' process occurring below 10cm and DL for the 'Levy' body motion process occurring above 10cm.The research found that DL on its own was insufficient for distinguishing Pollocks from non-Pollocks [17].Because our machine focuses on sizes of 10cm and larger, we would therefore expect to see little correlation between PMF and DL.When we employ DBC and SDBC fractal analysis techniques to quantify the scaling properties of the luminance scaling properties of the artworks we indeed found little correlation between PMF and their fractal dimensions (0.05).However, it's important to note that DBC and SDBC are just 2 possible approaches to measuring the fractal dimensions of art works, and future research should focus on a more comprehensive study of the relationship between PMF and these scaling parameters.
To explore Pollock's fractal pourings further, we compare them to computer-generated fractal images.Fig 19 demonstrates the visual similarity between the multi-scaled structure of Pollock's Number 32 and that of a fractal pattern generated using a midpoint displacement method [75].The Map of the fractal image is also shown in Fig 19 and is constructed using the multi-scaled tile technique described for the poured paintings in the Methods section.This Map reveals large regions of green that darken in the black and white regions of the fractal,  Given the prevalence of fractals in natural scenery, in Fig 21 we present a preliminary comparison of Pollock's One (1950) with 2 common natural images (a thicket and trees).All 3 images display the multi-scaled structure associated with fractal patterns.Intriguingly, the thicket is quantified by PMF = 0.71, One is quantified by PMF = 1, while the trees are quantified by PMF = 0.52.One possible explanation relates to differences in the fractal boundary density.In particular, the trees feature noticeably larger gaps devoid of the fractal boundaries.However, this result is also a useful reminder that although fractal analysis plays an important role, the machine is more than a simple fractal detector and many other visual characteristics will be in play.Although all 3 images are fractal, the natural images will differ in many subtle ways.A future investigation will perform a systematic study of a comprehensive set of natural images with the aim of using PMF to explore the connection between Pollock's and nature's fractal patterns.
Finally, we compare Pollock's images to a set of standard non-fractal designs to confirm that these are quantified by PMF = 0.These patterns encompass solid colors, bars, random pixels, gradients, and the distinctive 'Pick up sticks' configuration (which consisted of randomly oriented black lines against a white background).Examples are shown in Fig 22 .Our model demonstrates remarkable accuracy in classifying each of the test patterns as being a non-Pollock-with one intriguing exception involving a specific orientation (a 5-degree angle from the horizontal axis) and widths of black and white bars.The vulnerability to a specific line angle is surprising given that Pollock's all-over style is celebrated for being insensitive to painting orientation.Future research will examine trajectory orientation of Pollock's work in more detail to determine if there are stand-out orientations of individual features that 'average out' when combined into the dense interacting web of the all-over style.Interestingly, we note that the 8 poles within Blue Poles are much steeper than the 15-degree condition that challenged our model.

Image classifications and misclassifications: Example cases
The Pollock Dial shows that all but 2 non-Pollocks lie safely below PMF = 0.56, generating our model's high MA for detecting the visual miss-match of non-Pollock works.This is impressive given that our study features a diverse set of 284 poured works created by other artists.This set spans from images taken from Pollock's studio floor (as noted earlier, this is categorized as a non-Pollock in training because it lacks his compositional approach) to the poured creations of a Monkey.
A number of the non-  The inference set also features an image generated by a wind-driven pendulum built to harness fractal wind gusts to generate Pollock-like fractal patterns (Fig 23) [14].Despite this intriguing method of capturing nature's dynamic patterns, our machine is able to classify the resulting painting correctly as a 'non-Pollock' (PMF = 0.11).Finally, we also use paintings from Dripfests-the experiments aimed at exploring whether children's poured paintings have more visual similarities than the equivalent adult paintings to Pollocks [17,31].This hypothesis is based on proposed similarities between the body motions of Pollock and the children.Fig 4 shows example artworks by an adult and a child.Our machine is able to classify both the children's and adults' art works correctly as 'non-Pollocks'.Furthermore, the PMFs for the children (mean PMF = 0.002 ± 0.002) and adult (mean PMF = 0.009 ±0.007) paintings aren't significantly different between the 2 groups.
The Pollockizer.Our model misclassifies One generated by the 'Pollockizer'-a mechanical device developed to generate Pollock imitations (Fig 24) [35].This device consists of a container of paint that swings on a string, dripping paint onto a horizontal canvas positioned below.Based on the principle of a chaotic pendulum, the string can be knocked (either mechanically or via magnets) at close to the resonant frequency of the swinging motion.In doing so, fractal patterns are generated in the container's motion and therefore also in the paint trajectories recorded by the canvas below.Our machine suggests that, by generating a painting with PMF = 0.67, the Pollockizer is indeed capable of generating Pollock's artistic signatures.
AI-generated art.Given the growing prevalence of machine-generated art (e.g.those generated by DALL-E), a crucial question concerns whether our AI technique can detect AI-generated Pollock imitations.We test 87 'AI Pollocks' and classify them correctly with 100.00% accuracy.These consist of 3 groups of AI-generated Pollocks.DALL-E takes a textual description as an input and generates an image that corresponds to that description.We use the textual description "jackson pollock' and "jackson pollock imitation" to generate images at 1024x1024 pixels and these are tested for 4 different physical dimensions (25x25cm, 50x50cm, 100x100cm, and 200x200cm).Neural Love is a web-based tool that uses AI to create artistic images, avatars, and realistic portraits.We locate on-line images from Neural Love that are proposed to be imitation Pollocks.Their image sizes of 512 x 512 pixels are run through our machine at the same physical dimensions used for the DALL-E paintings.Although an algorithmic approach rather than strictly an AI approach, we also use Pollock Master images.This is a public github repository that makes Pollock-like images.The images are 5528 × 3572 pixels

Future models
We did not augment the image set in our current model.In machine learning, image augmentations are a set of techniques used to artificially increase the diversity of a data set by applying various transformations to the original images.These transformations can include image rotations, flips, scaling, changes in brightness and contrast, and more.The primary goal of image augmentations is to enhance the model's ability to generalize patterns from the training data to unseen examples.By introducing variations in the data set, the model becomes more robust and less prone to overfitting, which is known as generalization.Image augmentations are particularly valuable in computer vision tasks such as image classification when having a diverse and representative training data set can significantly improve model performance.
We undertake brief investigations of some additional models that utilize training augmentation but a more thorough investigation is needed beyond the scope of our current work.Below, we discuss 2 additional models that we train.Neither model out-performs our current model in MA nor by minimizing the PMF difference between images of the same painting taken from the 2 Pollock image groups.However, with the appropriate development, we anticipate that a future model with some augmentations could result in a model that is more generalizable than our current one while potentially maintaining our high MA performance.
Alternate Model 1 (MA: 97.7%).Augmentations: 50% of the images are converted to grayscale, flipped horizontally, flipped vertically, or rotated in steps of 90 degrees.There are several advantages associated with these augmentations: 1) Introducing flips, mirrors, and rotations removes any directional bias from all paintings.This would be helpful if the correct orientation of a painting is not known.2) Reducing the color influence forces the model to focus on the spatial patterns generated by the pouring process.This equalizes different image sources, some of which are grayscale and others are color.However, these augmentations introduce the following disadvantages which reduce the MA: 1) The correct orientations of Pollock paintings are known.This is therefore a real property of his work that we shouldn't eliminate.Instead, our current model simply rotates a painting of unknown origin and calculates each PMF to allow for alternative orientation options.2) Color can be an important characteristic of Pollock paintings and so reducing the presence of this characteristic diminishes the model's ability to distinguish.
Alternate Model 2 (MA: 97.1%).Augmentations: 50% of the images have their brightness and contrast values randomly adjusted between the values of -0.2 to +0.2 (using the Albumentations Python Library).These augmentations increase generalizability to various photographic and lighting conditions.The brightness and contrast plots vs PMF become much more stable than those of the current model (see Fig 26).However, these augmentations have the disadvantage that low contrast and brightness could be a real feature of Pollock's works and this augmentation therefore inappropriately reduces the MA.
In summary, augmentations have the potential to improve future models provided any challenges to computational resources are overcome.Lacking augmentations, our current model has a high MA and is robust to image variations through careful selection of image parameters.In particular, our model should only be run on paintings of unknown origin that have color images, limited image distortions such as printing artifacts, and limited brightness and contrast distortions.

Conclusion
We have developed a machine learning strategy that employs a novel image ingestion approach and that leverages the power of transfer learning.Our approach distinguishes between authentic and imitation Pollock poured works with an accuracy of 98.9% despite the limited number of images available for training.Based on this accuracy, we anticipate that our technique will be useful when combined with more traditional approaches to authenticity studies, in particular visual inspections by connoisseurs.Our generalization tests show that our machine performs well for artists who use the pouring technique.However, we emphasize that our current model is only applicable to poured paintings.Our PMF Threshold is termed a 'necessary but not sufficient condition' for image classification tasks.As an example, the high PMFs achieved by the computer-generated fractals are useful indicators of the machine's decision-making process, but they do not challenge the machine's usefulness as an authenticity tool-because they are not poured works.Further generalization studies will be needed to determine the extent to which our machine can be broadened to other artists and other artistic techniques.It is likely that when trained appropriately our machine will perform well when confronted with the visual complexity generated by other forms of gestural art.As AI examinations of art become more widespread, it will be important to understand the limitations of their applicability.For example, techniques based on facial recognition [79] will not be expected to perform equally well when faced with Pollock's complexity.Authenticity studies will most likely benefit from the employment of diverse machine models.
We have also shown that our AI approach can contribute to Pollock studies in ways that move beyond authenticity studies.To counter the black box nature of our approach, we developed Pollock Maps and Pollock Zoom-in Charts to probe the spatial and scaling signatures of Pollocks.When coupled with their quantifying parameters, these novel visual aids provide an interdisciplinary bridge between the machine's output and traditional art theory investigations of Pollock's work.In this way, AI has the potential to provide a new 'eye' on Pollock's all-over style.
Using these interpretational methods to look into the black box, our results indicate that the machine is examining the scale invariance of the poured patterns in its quest to distinguish between the Pollock masterworks and the imitations.A number of research groups have previously used various forms of fractal analysis to investigate Pollock's work, and it has been successfully employed in 2 high profile authenticity cases.However, researchers selected their techniques in these previous cases-introducing an element of subjectivity to the process.Here, the machine learned the usefulness of fractal analysis in an objective process beyond human influence.This objective approach will be useful not only for future comparisons of Pollocks with imitations but also with comparisons of Pollocks with natural images.Pollock famously declared "I am nature" to the art world and now science has the tools to confirm this connection between his fractals and those of nature using the machine's PMF values.Intriguingly, recent environment psychology research models the eye as a sophisticated fractal detector [43].Perhaps Pollock used his fractal eye to spot fractals in nature, the Pollock experts use their fractal eye to spot fractals in Pollock's work, and now the machine can do the same in a more quantifiable manner.
Finally, allowing AI to 'view' Pollock's art represents a major step in art appreciation and represents the latest step in the technological story set in motion by Taylor's first use of computer analysis of artworks [13].Writing about Taylor's computer analysis at the time, MOMA's chief conservator Jim Coddington declared: "In the visual arts we are at the beginnings of such a field and make no mistake, it is coming."[29].In the intervening years, the concept of AI has changed from science fiction to science fact and its arrival in the world of art will have many fascinating repercussions.

Fig 1 .
Fig 1. 81 x 110cm sections of the real (left) and replica (right) Blue Poles: Number 11, 1952 (210 x 486.8cm,National Gallery of Australia).https://doi.org/10.1371/journal.pone.0302962.g001 machine then averages the Pollock Signature across all of the tiles (i.e.across different locations and different sizes) to generate a signature for the whole artwork called the Pollock Matching Factor (PMF).The PMF value lies on a scale from 0 to 1.A PMF of 1 indicates that every tile in the work displays a Pollock Signature of 1.Similarly, a PMF of 0 indicates that all tiles in the work have a Pollock Signature of 0. Increasing values above 0 indicate that more and more of the artwork's tiles have a high (~1) Pollock Signature.A data flow diagram of the full training process is shown in Fig 2.

Fig 2 .
Fig 2. A flow diagram of the image pre-processing, data partitioning, and machine learning.The image is cropped and then tiled at multiple size scales (3 of the 22 tile scales are shown).The machine trains on the full set of tiles.During inference, the machine assigns a Pollock Signature to each of the tiles, which are then grouped by size and an average Pollock Signature is assigned.Finally, the average Pollock Signature is averaged across tile sizes to calculate the PMF.The intermediate numbers shown in the diagram are Pollock Signatures of the tiles and the final number is the PMF.https://doi.org/10.1371/journal.pone.0302962.g002 our model selection process, we investigate numerous deep learning model architectures (shown in Fig 3 and detailed in S3 Table).For each model test, we adjust the PMF Threshold to maximize the MA.Fig 3 illustrates the mean MA for various CNN model architectures, all of which use our multi-scaled tiling ingestion method (blue).The highest performing model within the ResNet50 model architecture is plotted separately (green) because ultimately this becomes our chosen model.The figure also includes 3 ViT model architectures that use our tiling method (orange) and 3 ViT model architectures that are trained only on the full-scale image (red).We train ViTs in these 2 different ways to better compare and understand the

Fig 3 .
Fig 3. Mean MA for the various model architectures considered.The blue bars represent CNN model architectures, including ResNet50.The orange/red bars represent ViT model architectures.Within each model architecture, the error bars quantify the variability in MA performance between models generated by choosing different random seeds during training.The green bar represents the highest performing model within the ResNet50 model architecture and ultimately becomes our chosen model.All models are trained using our multi-scaled tiling method, except for the red group models which are trained on the full images.While the red bars are trained using a batch size of 4 due to computational limitations, the faded red bar is trained with a batch size of 32.https://doi.org/10.1371/journal.pone.0302962.g003 Fig 3 we chose a batch size of 64, 1 epoch, and an image size of 256 x 256 pixels and 3 color channels.The orange ViT architecture models use an image size of 224 x 224 x 3 (necessary to match the input architecture), but otherwise use the same hyperparameters as the CNNs.Additional model parameters are outlined in S2 Table

Fig 4 .
Fig 4.An example of an adult (top) and child (bottom) Dripfest image (61.5 x 91.5 cm).https://doi.org/10.1371/journal.pone.0302962.g004 The Pollock Dial is shown in Fig 5.The green dots represent Pollock poured works and the red dots represent artworks created by the other artists.Representative images of varying PMF values are shown in the outer circle for comparison.Moving clockwise around the Dial, the titles of these artworks are: Untitled (Sam Francis), Adult 15 (Dripfest), Untitled (Wind Machine), Abstraction Orange (Jean-Paul Riopelle), Picasso's Guernica in the Style of Jackson Pollock (Michael Baldwin), Water Birds

Fig 9 .
Fig 9. Average PMF plotted against the image contrast (top) and image brightness (bottom) for the Pollock (green) and non-Pollock images (red).The broad colored regions correspond to the standard deviations in the data.The horizontal line represents the PMF Threshold.In each case, 0 corresponds to the undistorted value on the x axis.https://doi.org/10.1371/journal.pone.0302962.g009

Fig 10 .Fig 11 .
Fig 10.Zoom-in Chart for Dummy (see text for details).The images above the Chart show example tile images.https://doi.org/10.1371/journal.pone.0302962.g010 Fig 17 plots their relationship with PMF directly.The colored backgrounds are used to highlight the PMF ranges that are most populated by the artworks, and the dashed horizontal lines denote the mean values of SI, M, U, and C. A medium strength correlation can be seen between PMF and the spatial characteristics (U, C) of the artworks, consistent with Pollock's all-over style.The low

Fig 13 .
Fig 13.Spatial investigation of Dummy's Signatures.Grayscale image of Dummy (top left panel) and the same image with the Signature Map overlaid (right panel).The bottom panel focuses on Dummy's Map.The panel above the Map plots the average Pollock Signature as we move from left to right across the canvas and the panel to the right does the same as we move from bottom to top.https://doi.org/10.1371/journal.pone.0302962.g013

Fig 14 .
Fig 14.Spatial investigation of Blue Poles's Signatures.Grayscale image of Blue Poles (top left panel) and the same image with the Signature Map overlaid (right panel).The bottom panel focuses on Blue Poles's Map.The panel above the Map plots the average Pollock Signature as we move from left to right across the canvas and the panel to the right does the same as we move from bottom to top.https://doi.org/10.1371/journal.pone.0302962.g014

Fig 15 .
Fig 15.Spatial investigation of Pollock's Untitled: Cut-Out (77.3 x 57cm, Ohara Museum of Art, Japan).Grayscale image of Cut-out (top left panel) and the same image with the Signature Map overlaid (right panel).The bottom panel focuses on Cut-Out's Map.The panel above the Map plots the average Pollock Signature as we move from left to right across the canvas and the panel to the right does the same as we move from bottom to top.https://doi.org/10.1371/journal.pone.0302962.g015

Fig 16 .
Fig 16.A timeline chart of the evolution of the various machine parameters (PMF, SI, M, C, and U) along with other artwork parameters (N, A, and AR) spanning from 1943 to 1954.See text for details.https://doi.org/10.1371/journal.pone.0302962.g016

Fig 18 .
Fig 18. Radial maps for Pollock's Water Birds (66.4 x 53.8cm, Baltimore Museum of Art, USA) (top row), Pollock's Blue Poles (middle row), and Henri Michaux's Untitled (74.9 x 107.9cm, Edward Thorp Gallery, New York) (bottom row).The left column shows the artwork, the middle column shows the radial map plotted using the absolute colors, and the right column shows the radial map plotted using relative colors.The top and bottom bars show the relative and absolute color ranges, respectively.See text for details.https://doi.org/10.1371/journal.pone.0302962.g018

Fig 19 .Fig 20 .
Fig 19.Left: A 63.5 x 63.5cm section of Pollock's Number 32, 1950 (269 x 457.5cm,Kunstsammlung Nordrhein-Westfalen, Germany).Middle-left: A 100 x 100cm computer generated fractal image.Middle-right: the Map of the fractal image.Right: the Map constructed from 10cm tiles (this tile size is shown in the middle left image as a yellow square).https://doi.org/10.1371/journal.pone.0302962.g019 Pollock poured paintings are by other famous artists.Pollock's contemporary Max Ernst features in the training (Young Man Intrigued by the Flight of a Non-Euclidean Fly from 1942-47) and inference (The Bewildered Planet from 1942) sets.Works by Abstract Expressionist contemporaries Hans Hofmann (Fantasia from 1943 and The Wind from 1944) and Arshile Gorky (One Year The Milkweed from 1944) feature in the training set.Artists Marcel Barbeau and Jean-Paul Riopelle from the French Canadian movement Les Automatists, which had a similar artistic mission to the American Abstract Expressionism movement, feature in the training and inference sets.From the 1960s, Niki de Saint Phaelle's art created by shooting at balloons filled with paint (Shooting Picture 1 and 2 from 1961) features in the training set, along with Norman Rockwell's famous parody of Pollock's work (The Connoisseur from 1962).Four of Michael Badwin's 1980s series (Pollock-inspired poured paintings featuring embedded images) appear in the training set and 2 appear in the Inference set.Sam Francis's Untitled from 1985 appears in the inference set.Eight of Prince Jurgen von

Fig 26 .
Fig 26.Average PMF plotted against the image contrast (top) and image brightness (bottom) for the Pollock (green) and non-Pollock images (red).The broad colored regions correspond to the standard deviations in the data.The horizontal line represents the PMF Threshold.In each case, 0 corresponds to the undistorted value on the x axis.https://doi.org/10.1371/journal.pone.0302962.g026

Table 1 . A selection of AI techniques listed in chronological order. Reference Approach Data Set Measure Result
[50](AlexNet) to the Rijksmuseum data set and demonstrated that CNNs could be effective at art authentication.Subsequently, van Noord et al expanded their approach and achieved a higher classification accuracy on the same data set by adopting a multi-scale CNN[50].Dobbs and Ras have achieved the highest MA so far on the Rijksmuseum data set by using Resnet101.
Table provides a comprehensive list of the image sets.The image sets feature 2 overall categories of artwork-those established as being created by Pollock and those established to be by other artists.

Table 2 . Data partitioning of Pollock and non-Pollocks according to the number of artworks, the number of images of these artworks, and the number of tiles in these images.
https://doi.org/10.1371/journal.pone.0302962.t002